assessingthemarketvalueofrealestate ... reprints... · 2008 v36 4: pp. 717–751 real estate...

2008 V36 4: pp. 717–751

REAL ESTATE

ECONOMICS

Assessing the Market Value of Real EstateProperty with a Geographically WeightedStochastic Frontier ModelStephen A. Samaha∗ and Wagner A. Kamakura∗∗

In this study we consider the problem of sellers, buyers and real estate apprais-ers in determining the price for a house, taking into account the characteristicsof the house and its location as well as the goals of these three different parties.The appraiser’s job is to determine the fair market value of the house, whilethe buyer and seller want to find, respectively, the lowest and highest feasibleprice for it. We combine recent developments in geography and economet-rics to develop an approach that determines local estimates of property valuesfrom the perspectives of the buyer, seller and appraiser, taking into account thecharacteristics of the house as well as its location. We illustrate our approachanalyzing closing prices in one residential real estate market.

In this study we propose to take the perspectives of the seller and the buyer inuncovering the lowest price that the seller should accept or the highest price thebuyer should pay for the real estate property, while simultaneously factoring inthe critical aspects of market imperfections and location. Because real estatetransaction prices are often settled in a less-than-perfect negotiation process,and, consequently, the estimates of property values based on past prices shouldleave room for negotiation, we develop a geographically weighted stochasticfrontier model that takes the seller’s perspective of determining the potentialbuyer’s reservation value (i.e., the highest possible price she should expectfor the house) as well as the buyer’s perspective of determining the seller’sreservation value (i.e., the lowest feasible price to bid on a house). We thenapply and illustrate our model using data from one real estate market.

It is difficult to overstate the central role that the hedonic price regressionframework has had in valuing residential and commercial real estate proper-ties. This framework (Griliches 1961, Rosen 1974, Epple 1987) assumes thatthe prices sold for each property represent the market clearing prices, andtherefore the results of the regression should provide an unbiased estimatefor the fair market value of each house. However, there is an increasing body

∗University of Washington, Seattle, WA 98195 or [email protected].∗∗Duke University, Durham, NC 27708 or [email protected].

C© 2008 American Real Estate and Urban Economics Association

718 Samaha and Kamakura

of research that questions whether these core assumptions of perfect marketequilibrium truly hold in an empirical setting. It is now known that real estateproperties can suffer from numerous market imperfections. These imperfec-tions or inefficiencies can result from many sources, including transactionnoise resulting from the imperfect matching of buyers and sellers, idiosyn-cratic bargaining outcomes and unique physical asset characteristics that implyan incomplete market in comparables (Childs, Ott and Riddiough 2002). Inaddition, imperfections can also occur because of liquidity, private demands ofinvestors and bilateral bargaining outcomes (Quan and Quigley 1991). Hard-ing, Knight and Sirmans (2003) also find that household wealth, gender andother demographic traits influence bargaining power and therefore negotiatedprices.

Research has also shown that sellers are heterogeneous in their motivationsto sell, and these motivations can also affect sales prices. Glower, Haurin andHendershott (1998) find that home sellers who are motivated to sell morequickly will set a lower list price, have a lower reservation price and acceptearlier, lower offers. Quan and Quigley (1991) present a model of heterogeneousbuyers and sellers and conclude that the greater a seller’s urgency is to sell, thelower the seller’s reservation price. Miceli (1989) also notes that sellers maynot want to simply maximize sale price but may also consider “minimizingthe amount of time it takes to obtain a pre-agreed price.” Geltner, Kluger andMiller (1991) discuss how variations in seller holding costs also affect theseller’s reservation price and find that the greater the time cost of holding aproperty, the lower the seller’s reservation price.

Because of these inefficiencies, the analyst has highly imperfect informationabout the seller’s and buyer’s reservation values; the analyst only knows thatthe observed transaction price is equal to or greater than the seller’s reservationvalue and equal to or lower than the buyer’s reservation value. Because thefactors affecting the final transaction price are rarely observed, it is imperativeto not only model them as errors in the hedonic setting, but also to accountfor these unobservable imperfections in estimating the buyer’s and seller’sreservation values. Sales efficiencies could be improved if buyers and sellershad not only a best guess of a property’s true value, but also a sense for howidiosyncrasies in the real estate transaction process can affect the negotiationprocess and subsequent price range that the home could conceivably be sold for.This range can be modeled as stochastic due to the stochastic nature of theseerrors, and, indeed, Glower et al. (1998) hypothesize that variations in listprices of similar properties are attributed to random seller errors and explicitlyintroduce stochastic errors into list price as the sum of the optimal price andthe seller’s error in setting the list price.

Weighted Stochastic Frontier Model 719

However, it is not sufficient to simply incorporate these phenomena into thehedonic framework without also factoring in the critical aspects of location andthe role that geographic location plays in valuing properties. It is well knownthat there are many unobservable characteristics that can affect market value:proximity to amenities (waterfront, mountain view) and disamenities (railroad,highways, power lines, telecom towers, traffic conditions and adjacent negativeland uses) are but a few examples. Some of these unobservable factors aredirectly related to geographic location, while others, such as curb appeal, adderrors to the model and also uncertainty in the valuation of the property bybuyers and sellers. It is critical to be able to explicitly model these unobservablefactors as well to allow for model realism.

The Efficiency of Real Estate Markets

Because hedonic price regressions assume that transaction prices for each prop-erty represent the market clearing prices, the results of the regression shouldprovide an unbiased estimate for the fair market value of each house, underthe perfect-market assumption. However, as stated earlier, there are certaininefficiencies present. For example, the set of houses available for purchaseat any given time is limited, and it does not cover the entire characteristicsspace, giving each house some monopoly over those buyers whose equilibrialie in its vicinity in the characteristics space (Rosen 1974). Second, despitetechnological advances in the multiple listing service, search costs are still highfor buyers, forcing them to make trade-offs between their search costs and themarginal benefit of efficient buying decisions (Ratchford 1980). Third, partic-ipants in real estate markets often have incomplete information regarding thehome’s attributes, and decisions to buy and sell must often be made on partialknowledge. Trades are also decentralized and market prices are the outcomesof pair-wise negotiations (Quan and Quigley 1991). These inefficiencies canconsequently lead buyers and sellers to respectively pay more for and acceptless than their respective optimal prices. Fourth, market inefficiencies may alsoresult because of potential conflicts of interests between brokers and sellers.Although sellers hope that properties will sell at the highest possible prices,brokers have an incentive to sell the house quickly. In general, the lower theasking price, the lower the effort required on the part of the broker to sell theproperty and the easier it is for the property to sell (Geltner et al. 1991). Brokersare also heterogeneous with respect to their ability levels, which may enter intothe determination of both selling price and time on the market (Miceli 1989,Miller 1978). Last, bargaining power can also affect the outcomes of real estatetransactions. Previous research has found systematic evidence in the housingmarkets that weak buyers pay higher prices and weak sellers receive lowerprices for their homes (Harding et al. 2003, Harding, Rosenthal and Sirmans2003, Colwell and Munneke 2006).


Given these uncertainties and inherent market inefficiencies, the observed trans-action price does not necessarily represent the equilibrium between the seller’sand buyer’s valuations. In an imperfect market such as real estate, the trans-action price is only known to be higher (lower) than or equal to the seller’s(buyer’s) reservation value. Therefore, individual buyers and sellers would bebetter informed with valuations at the Pareto frontiers rather than at the least-squares line as defined by traditional hedonic price regression models. In otherwords, in order to obtain a better sense of potential buyers’ valuations, theseller wants to estimate the buyer’s hedonic price at the frontier, taking intoaccount that the buyer’s reservation value is at least as high as or higher thanthe observed transaction price. Similarly, when deciding on an offer of whetheror not to buy a house, the buyer wants to estimate the seller’s hedonic price,taking into account that sellers’ reservation values in the past were at least aslow as or lower than the observed transaction prices.

Stochastic frontier estimation originated under the contexts of production eco-nomics with the work done by Meeusen and van den Broeck (1977), Aigner,Lovell and Schmidt (1977) and Battese and Corra (1977). Conventional econo-metric production models treated producers as successful optimizers and uti-lized methods such as those from Cobb and Douglas (1928) to estimate pro-duction, cost and profit functions. Under this classic framework, deviationsfrom the maximum possible output or profit or minimum cost, given a set ofinputs, were attributed exclusively to random statistical noise. Stochastic fron-tier estimation was developed to incorporate a theory of producer behavior thatexplicitly incorporated the possibility of suboptimal performance in additionto random statistical noise. The efficient frontier under this context (from thestandpoint of a producer, for example) would represent the maximum outputthat is possible for a given set of inputs and technology. Producers who are op-erating on this frontier are efficient, while those operating beneath this frontierare technically inefficient. In our application to hedonic pricing, the stochasticfrontier represents the maximum price that the home could be sold for (i.e.,the buyer’s reservation value) given its measurable characteristics. Similarly,the inverse stochastic frontier represents the minimum price that the homecould conceivably be sold for (the seller’s reservation value). For more detailsregarding stochastic frontier estimation, see Kumbhakar and Lovell (2000).

Accounting for Geographic Heterogeneity in Hedonic Price Analysis

Because properties near one another tend to be influenced by similar factors(size, age, architectural type and design, proximity to schools and city cen-ters, etc.), the selling prices of these houses may be correlated over geography.There have been numerous attempts to incorporate spatial characteristics and


correlation into hedonic models. Can (1992) and Can and Megbolugbe (1997)have allowed the marginal prices of property attributes to vary according toneighborhood characteristics. Similarly, Day (2003), Day, Bateman and Lake(2003) and Goodman and Thibodeau (2003) sought to separate hedonic pricefunctions into sets of socioeconomically homogeneous neighborhoods. Theidea of using submarkets (areas in which the prices or the quantities of at-tributes differ from those in different areas) to address spatial autocorrelationin the data is not new. Wilhelmsson (2004) used cluster analysis to definefunctional submarkets as dummy variables to address the problem of spatialautocorrelation, while Dale-Johnson (1982) used factor analysis to define sub-markets for hedonic pricing. Goodman and Thibodeau (1998) used hierarchicalmethods to define submarkets, while Bourassa et al. (1999) used a combinationof principal components with cluster analysis.

Some researchers have also incorporated polynomial expressions of latitudeand longitude coordinates into their models (Dubin 1992, Pace and Gilley1997, Pavlov 2000, Clapp 2001) or the interactions of the latitude and longi-tude coordinates (Fik, Ling and Mulligan 2003) to model spatial error depen-dence. Still other approaches have included location value signature models(Fik et al. 2003), local polynomial regressions (LPR) (Host 1999), LPR withBayesian smoothing (Clapp, Kim and Gelfand 2002) and smooth-spatial ef-fects estimators (Gibbons and Machin 2001, 2003). Others have approachedthe problem by implanting simultaneous autoregression and simultaneous tem-poral autoregression (Pace 1997, Pace and Barry 1997, Pace et al. 1998), aswell as conditional autoregression, Kriging (Dubin 1992, Basu and Thibodeau1998, Dubin, Pace and Thibodeau 1999) and spatial error dependence models(Kelejian and Prucha 1999, Bell and Bockstael 2000). Last, geographicallyweighted regression (GWR) (Fotheringham, Brunsdon and Charlton 2002) andspace-varying regression coefficients (Pavlov 2000) have been used to analyzethe spatially varying relationships of housing characteristics and prices acrossgeography.

In order to account for the inefficiencies in real estate markets and the fact thatreal estate values are location dependent, we next propose a new model thatcombines the concept of geographically weighted estimation with stochasticfrontier estimation. This new model allows us to assess the potential marketvalue of homes at the Pareto frontiers, both from the perspective of the sellertrying to find the highest (buyer reservation) value for the home and for thebuyer seeking to find the lowest possible (seller reservation) value to pay. Thedifferences between the unobservable buyer and seller reservation values definethe range or latitude for negotiation between the two parties.


Assessing Property Values at the Pareto Frontier

As mentioned earlier, our goal is to extend the traditional hedonic price regres-sion framework for real estate assessment in two ways. First, we acknowledgethat real estate markets are imperfect and that the observed transaction pricesare higher (lower) than the seller’s (buyer’s) unobserved reservation values.Therefore, we attempt to obtain better estimates of the seller’s and buyer’sreservation values by fitting the hedonic price regression at the Pareto frontiers,both from the perspective of the buyer attempting to find the lowest possibleprice to pay or the seller seeking the highest price that the market can bear. Sec-ond, we take into account that the physical features of a house may be valueddifferently depending on its location; buyers looking for an urban residencemay attach different value for the features of a house than those interested insuburban or rural living.

A Pareto price frontier (Kumbhakar and Lovell 2000) can be written as

yi = f (xi ; β) · TEi , (1)

where yi represents the transaction price for home i (i = 1 . . . n), xi is the vectorof physical characteristics for that home, β is the vector of parameters to beestimated, f (xi; β) is the Pareto (or maximum price) frontier and TEi definesthe technical efficiency (the ratio of the observed price that the home sold forto the maximum feasible price). Thus, the observed selling price of a home, yi,reaches its maximum feasible value of f (xi; β) only when TEi = 1. Otherwise,per Kumbhakar and Lovell (2000), TEi < 1 provides a measure of the shortfallof observed price from the maximum feasible price.

In order to augment the above deterministic frontier to incorporate house-specific random shocks, the above specification can be rewritten as

yi = f (xi ; β) · exp{vi} · TEi , (2)

wheref (xi; β) · exp {vi} is the stochastic frontier in which f (xi; β) representsthe deterministic portion common to all homes being sold and exp {vi} repre-sents the random shocks on each particular home. Assuming that f (xi; β) takesthe log-linear Cobb-Douglas form, and letting TEi = Exp{–ui}, the stochasticfrontier can be rewritten (Aigner et al. 1977) as

ln yi = β1 +∑

k

βk ln xik − |ui | + vi, (3)

where u ∼ N(0, σ 2u), that is, distributed as nonnegative half normal and

v ∼ N(0, σ 2v ), where ui represents the inefficiency term and v i represents

the idiosyncratic effects specific to each house. Then, letting ε i = v i – |ui|, δ =σ u /σv , σ = (σ 2

u + σ 2v )1/2, and assuming the half-normal model, the log-density

(Aigner et al. 1977) can be written as


ln hi = − ln σ − 1

2ln

2

π− 1

2

(yi − x ′

iβ

σ

)2

+ ln �

(−δ(yi − x ′iβ)

σ

), (4)

where �(z) is the cumulative distribution function of the standard normaldistribution. Then, a local maximum likelihood function (Fan, Farmen andGijbels 1998) for a focal location j can be expressed as

ln Lj =∑i �=j

Wij · ln hi, (5)

and the local estimator is

θ̂j = arg maxθ

∑i �=j

Wij · ln hi, (6)

where

• θ̂j represents the vector of local parameters to be estimated by localmaximum likelihood; that is, θ = {β 1 . . . β k, δ, σ},

• Wij = Exp[−λ21(zj1 − zi1)2 − λ2

2(zj2 − zi2)2] are the geographicweights,

• zi1 = longitude coordinate for location i,

• zi2 = latitude coordinate for location i and

• λ1, λ2 are the longitudinal and latitudinal weighting parameters to beestimated.

Equation (5) shows the log-likelihood function of a half-normal stochasticfrontier regression (Kumbhakar and Lovell 2000), except that it is estimatedat a given location j. The geographic weights Wij define how relevant the datafrom a neighbor i is, depending on its distance from the focal location j alongthe East–West (longitude) and North–South (latitude) directions. We allow thegeographic weights to decay differently in the two directions, leading to ananisotropic geographically weighted model, which depends on the parametersλ1, λ2 that are to be estimated.

Estimation of the two geographic decay coefficients λ1 and λ2 can be done bycross-validation (see Fotheringham et al. 2002 for a discussion of calibratingspatial weighting functions with cross-validation). This calibration of the decaycoefficients can be attained by solving the following nonlinear optimizationproblem:


θ̂ = arg maxθ

N∑j=1

∑i �=j

e−λ21(zj1−zi1)2−λ2

2(zj2−zi2)2

×[− ln σj − 1

2ln

2

π− 1

2

(yi − x ′

iβj

σj

)2

+ ln �

(−δj (yi − x ′iβj )

σj

)], (7)

where θ = {λ1, λ2, (βj1 . . . βjk, δj , σj ); j = 1, . . . , N}.

Once the geographic decay coefficients λ1 and λ2 are determined for the cali-bration sample of sold houses, the coefficients for the local stochastic frontierfor the buyer’s reservation value at any new location A can be obtained bymaximizing the log-likelihood below, which depends only on the data fromcalibration sample

θ̂Buyer RV

A = arg maxθ

∑i �=A

e−λ21(zA1−zi1)2−λ2

2(zA2−zi2)2

×[− ln σA − 1

2ln

2

π− 1

2

(yi − x ′

iβA

σA

)2

+ ln �

(−δA(yi − x ′iβA)

σA

) ], (8)

where θ = {βA1 . . . βAk, δA, σ A}.

Note that the global anisotropic parameters, {λ1, λ2}, which are estimated viacross-validation across the calibration sample of sold houses, are treated asconstants in this local maximum likelihood estimation.

Hence, the local frontier for the buyer’s reservation value is given by themaximization of the local log-likelihood function above, whereas the localfrontier for the seller’s reservation value is the negative half of the half-normaldistribution, which requires a substitution from the above term �(z) to [1 –�(z)]. Thus,

θ̂ Seller RVA = arg max

θ

∑i �=A

e−λ21(zA1−zi1)2−λ2

2(zA2−zi2)2

×[− ln σA − 1

2ln

2

π− 1

2

(yi − x ′

iβA

σA

)2

+ ln

(1 − �

(−δA(yi − x ′iβA)

σA

)) ]. (9)


Table 1 � Data summary statistics.

Variable Minimum Mean Median Maximum Std. Dev.

Total heated square feet 746 2,209.24 2,085 7,231 855.27Total unheated square feet 0 187.39 0 3,360 397.65Number of bedrooms 1 3.48 3 6 0.80Total acres 0 1.00 0.46 33.5 1.93Age of property (years) 0 17.58 13 92 16.58Number of bathrooms 1 2.60 2.5 7 0.80Number of garages 0 1.12 1 3 0.99Number of fireplaces 0 0.96 1 4 0.53Sold in spring 0 0.34 0 1 0.47Sold in summer 0 0.31 0 1 0.46Sold in fall 0 0.19 0 1 0.39Number of days on the 0 70.52 41 780 87.21

marketHouse is vacant 0 0.14 0 1 0.35Seller adjusted list price 0 0.28 0 1 0.45Seller granted concessions 0 0.14 0 1 0.34Seller requires appointment 0 0.66 1 1 0.47

Assessing the Value of Residential Real Estate

with Hedonic Price Analysis

In addition to estimating the geographically weighted stochastic frontiers, wealso estimate a geographically weighted hedonic model to be used as thebenchmark for fair market value. This will be useful for comparative purposesand interpretation later. Data were collected from a multiple listing service(MLS) database for a medium-sized county in the continental United States.The data collected included all detached single-family homes sold in that countyfor a particular year, their corresponding sales prices and predictor variablesbelieved to influence the prices of the homes, as well as their geographiccoordinates. A sample size of 1,035 homes was used in the study. We list belowthe variables used in our hedonic price analysis. These variables include boththe characteristics of the home as well as factors believed to reflect whether thebuyer was in a strong bargaining position at the time of sale. Summary statisticsfor them are reported in Table 1.

• Log selling price: The dependent variable, which is the natural loga-rithm of the closing price in dollars for the home.

• Log heated square feet: The natural logarithm of total heated squarefootage of the home.

• Log unheated square feet: The natural logarithm of total unheatedsquare footage of the home.


• Number of bedrooms: The total number of bedrooms in the home.

• Total acres: The total size of the lot in acres that the home sits on.

• Age of property: The age of the property in years.

• Number of bathrooms: The effective number of full bathrooms in thehouse. Note that half-bathrooms count as 0.5 full bathrooms. Hence, ahome with two full bathrooms and one half-bathroom has effectively2.5 full bathrooms.

• Number of garages: The total number of garage spaces.

• Number of fireplaces: The total number of fireplaces.

• Days on the market: The number of days the home was listed beforebeing sold.

• Closed in spring: A seasonal indicator variable that designates whetherthe home was sold in the spring quarter (April, May or June).

• Closed in summer: A seasonal indicator variable that designateswhether the home was sold in the summer quarter (July, August orSeptember).

• Closed in fall: A seasonal indicator variable that designates whether thehome was sold in the fall quarter (October, November or December).

• House is vacant: An indicator variable that designates whether thehome was vacant at the time of sale.

• Seller adjusted list price: An indicator variable that designates whetherthe list price at the time of sale was different from the original listingprice.

• Seller granted concessions: An indicator variable that designateswhether the seller at the time of sale granted any buyer concessionsnot directly related to the sales price of the home. Examples include theseller paying for all of the closing costs, agreements to make repairs orcoverage for any inspections and warranties.

• Seller requires appointment: An indicator variable that designateswhether the seller required a prior appointment for the prospectivebuyer to view the property.

Accounting for the Endogeneity Between Days on the Marketand Selling Price

The relationship between days on the market and selling price has been studiedquite extensively in the real estate literature (Miller 1978, Taylor 1999, Knight


2002, Anglin, Rutherford and Springer 2003 ). Under standard search theory,we should expect a positive relationship between the number of days on themarket and selling price, because, by waiting, a seller increases the probabilityof encountering a buyer with a high reservation price. On the other hand, homesthat remain unsold may become stigmatized in the eyes of potential buyerswho regard a long market duration as evidence of some defect. Taylor (1999)explains this phenomenon as “negative herding” and suggests that a longertime on market may result in a lower sales price. In addition, sellers with highsearch costs should be more impatient for a sale and thus more willing torevise a price downward if an early sale is not achieved (Knight 2002). Undereither theory, however, the days on the market and selling price are likely to bejointly determined; consequently, we use an instrumental variables approachvia a two-stage least squares regression to correct for simultaneity betweenthe selling price and days on the market, following an approach similar to thattaken by Harding et al. (2003). Specifically, we include four indicator variablesmeasuring atypicality: whether the list price is on the top and bottom deciles oflist prices in this market and whether the home is very new (less than 3 yearsold) or very old (more than 35 years old). We also decided to omit the two otherpotential indicators of atypicality used by Harding et al. (2003) (whether thehouse has unusually high numbers of bedrooms and bathrooms), as they weredetermined to be weak instruments for our particular sample.

In addition to atypicality, we also use five other indicators believed to influenceselling time on the market. These variables indicate whether the property beingsold was (1) in an urban or rural area, (2) located in one town deemed to bequite popular in the area, (3) had a dining room, (4) had a family room and (5)had vinyl siding. The results from the first stage ordinary least squares (OLS)regression are presented in Table 2.

As can be seen from the results, homes located in urban areas or in the populartown tend to sell faster. This may be reflective of the density level of potentialbuyers for properties in urban locations: properties situated in denser marketshave a faster exposure rate and shorter expected time to sale (Geltner et al.1991). The measures for atypicality also indicate that homes with very low orhigh prices or homes that are either very new or very old also take longer tosell. Hence, this is consistent with the findings of Haurin (1988) in that themarketing times of atypical houses should be relatively longer than those ofstandard houses. As expected, homes with dining rooms, family rooms andthose sided with vinyl also take longer to sell.

In the second stage, we substitute these predicted values as the explanatoryvariable for days on the market for our analysis below and proceeded withestimation.


Table 2 � First-stage regression results: Days on market.

Parameter

Variable Estimate t Value Pr > |t|Intercept 56.11 6.2 <.0001House is very new 42.57 5.1 <.0001House is very old 40.93 5.4 <.0001Listing price is very high 33.89 3.8 0.0002Listing price is very low 22.15 2.3 0.0231House is located in urban area −20.70 −2.9 0.0038House is located in desirable city −23.84 −3.0 0.003House has dining room 23.18 3.0 0.0028House has family room 13.32 2.4 0.0172House is sided with vinyl 12.22 1.4 0.1582

R2 0.113Adjusted R2 0.105

Testing for Functional Form Misspecifications and Structural Breaks

Before continuing, we also check our model for evidence of omitted variables,functional form misspecifications and structural breaks. As a first step, indica-tor variables were created for the number of bedrooms and bathrooms and runin place of the actual number of bedrooms and bathrooms. After examining theresults, the adjusted R2 remained unchanged (0.883) between the two specifi-cations (dummy coding vs. using actual numbers of bedrooms and bathrooms).In addition, many of the indicator variables in the dummy coding specifica-tion were found to not be statistically significant, and, at the same time, thisspecification required an additional 14 parameters to be estimated. Because theoriginal specification is more parsimonious and yields the same adjusted R2,we retain the original specification of using the actual numbers of bedroomsand bathrooms rather than using indicator variables.

In addition, we also tested for functional form misspecifications by employingan auxiliary regression to check for additional possible nonlinearities in thecovariates. Specifically, we use the Regression Equation Specification ErrorTest based on Ramsey (1969) to check for possible higher order terms in themodel (quadratic and cubic) for the given covariates. The results of the F testare nonsignificant: the p values are 0.728 for a quadratic functional form and0.386 for a cubic form. Hence, we do not reject the null hypothesis and retainour original functional form assuming linear predictors for subsequent analysis.

Last, because a nontrivial number of homes in our sample (15.5%) sold abovetheir respective list prices, we conducted a Chow test for structural change


(Chow 1960) to determine whether the regression coefficients differed acrossthe homes that sold above their respective list prices and those that did not. Wefailed to find a significant difference at the 5% significance level (p value =0.085). Hence, we failed to determine that a substantial difference exists. Wealso regressed the sales price against the usual housing characteristics but alsoincluded a dummy variable (1 = sold above list price, 0 = otherwise) which wasinteracted with our original covariates. None of the interactions were significantat the α = 0.05 level.

The Appraiser’s Perspective: The Fair Market Value

In order to arrive at the fair market value of a house at any given location, weapplied Fotheringham et al’s. (2002) GWR to our data, resulting in localizedestimates of the hedonic weights for various characteristics of the house. GWRis the equivalent of our geographically weighted stochastic frontier model in(8) with normal, rather than half-normal errors (i.e., setting δ = 0), althoughestimation is much simpler as it can be done through (geographically) weightedleast squares estimation. See Fotheringham et al. (2002) for details about theGWR model and its estimation. The first step in this process is to estimate theoptimal anisotropic spatial weighting function, which determines the relevanceof each observed data point for the hedonic price regression, based on howclose each observation is to the focal location.

The anisotropic spatial weighting function was calibrated using geographicallyweighted cross-validation (Cleveland 1979) leading to estimates of λ̂1 = 12.43and λ̂2 = 24.99, thereby indicating that the influence of neighboring locationson a particular home’s value decays more rapidly in the North–South than inthe East–West direction. In other words, neighboring homes that are offsetlongitudinally from a particular home have a correspondingly greater influencein determining the value of that home than those located the same distance awayin the North–South direction. This might be because the region we analyzed ismore elongated in the north–south direction, as shown in the map. Because thegeographic distribution of homes varies for each county and our data are basedon a cross-section of homes sold in a particular county at a particular time, ourresults do not necessarily generalize to other counties or to other time periods,which suggest the critical roles that distance, direction and time can play inlocal markets for defining and creating a list of market comparables to base anappraisal on.

A natural question to ask is whether the additional computational complexityrequired for GWR models actually results in significantly better predictiveperformance and fit over a standard OLS model. Our results show a higher cross-validated R2 value for the GWR model (0.908) than for the simple OLS model


(0.881). While at first glance the differences may not appear to be substantial,one must also take into account that they reflect predictive performance, ratherthan just mere fit.

Examination of the Geographically Weighted Parameter Estimates

We present a summary of the GWR parameter estimates along with those froma standard OLS model in Table 3. Because the GWR model produces oneset of parameter estimates for each location, Table 3 reports the means andstandard deviations for these estimates across all locations. The coefficientsfor Log Heated and Log Unheated Square Feet show the respective estimatedpercentage increases in price for a given percentage increase in heated and un-heated square footage, respectively, for the home. The fact that these estimatesare smaller than one show that the cost per square foot decreases with housesize. The intercept for the GWR model accounts for unobservable geographicheterogeneity in house prices not captured by the geographic changes in valu-ation for the physical characteristics of the houses. When looking at the OLSmodel, two noteworthy results are that age of property has a positive impacton price and that, as the number of days on the market increases, selling pricedecreases. As we shall see shortly, when the coefficients are allowed to varygeographically, the parameter estimate for number of days on the market variesfrom positive to negative, depending on location. The positive impact of age onprices may be due to other unobservable factors (such as quality of constructionor historical value) associated with age.

Examination of the Bargaining Position Variables

As mentioned earlier, we have also included a series of indicator variables toindicate the degree to which a buyer may have been in a strong bargainingposition at the time of sale.1 As illustrated in Table 3, homes that are vacant atthe time of sale can be expected to sell for less than homes that are occupied.This result seems to be consistent with previous research in that sellers ofvacant homes are in a weaker bargaining position than sellers whose homesare occupied (Harding et al. 2003). As noted by Harding et al. (2003), fullyfurnished homes are often more appealing to buyers than vacant homes, andthe sellers of vacant homes bear the full cost of carrying the home with nooffsetting benefits from occupancy or rental income. The fact that we allowour estimates to vary as a function of geographic space implies that the impact

1 We originally also interacted the indicators for bargaining position with the housingcharacteristics but found an overwhelming majority of these interactions to be statis-tically insignificant. Consequently, we proceeded with using only the main effects forthese indicator variables.


Table 3 � Parameter estimates from geographically weighted (GWR) and ordinaryleast squares (OLS) regressions.

Fit Statistics OLS Model GWR Model

Cross validated R2 0.881 0.908Full data R2 0.885 0.928Full data adjusted R2 0.883 0.927

GWR GWR GWROLS OLS Mean Parameter ParameterParameter t Parameter Std. Coeff.

Variable Name Est. Value Est. Dev. Variation

Intercept 5.40 28.7 5.95 0.458 0.077Log heated square feet 0.88 30.3 0.80 0.081 0.102Log unheated square

feet0.01 6.9 0.01 0.002 0.240

Number of bedrooms 0.001 0.1 0.005 0.013 2.610Total acres 0.01 4.8 0.03 0.012 0.444Age of property (years) 0.003 8.1 0.001 0.001 1.067Number of bathrooms 0.12 11.1 0.10 0.032 0.334Number of garages 0.04 5.0 0.04 0.013 0.334Number of fireplaces 0.10 8.3 0.08 0.018 0.234Sold in spring 0.02 1.4 0.03 0.022 0.734Sold in summer 0.04 2.7 0.04 0.014 0.375Sold in fall 0.01 0.3 0.01 0.026 2.789Number of days on the

market−0.0017 −8.6 0.0005 0.001 1.180

House is vacant −0.06 −3.5 −0.06 0.021 −0.347Seller adjusted list

price−0.03 −2.3 −0.03 0.024 −0.849

Seller grantedconcessions

−0.05 −3.5 −0.04 0.022 −0.490

Seller requiresappointment

−0.02 −1.4 −0.01 0.023 −3.129

Note: Bolded OLS estimates and t values are statistically significant at the 0.01 level.

of vacancy on selling price changes, depending on location. For example, ifa seller’s relative weakness is due to a high carrying cost but the demand forhomes in that location is high, then this higher demand could offset some ofthe weakness in bargaining position due to the home being vacant.

We also observe that sellers who adjust their list prices tend to sell their homesfor less than sellers who maintain the same list price throughout. One expla-nation may be that sellers with higher search costs should be more impatientfor a sale and therefore more willing to revise a list price if an early sale isnot achieved. Indeed, it has previously been established that such impatience


results in lower bargaining power (Binmore 1992). In addition, seller motiva-tions are likely to increase over time, as for example when a planned movingdate approaches or when an unmatched seller finds a desirable new home(Glower et al. 1998). Knight (2002) looks at the impact of changes to list priceand shows that homes whose list prices were revised took longer to sell andsold for less than homes that were initially priced correctly. Last, homes withlist price changes may have been mispriced initially and consequently mayhave been on the market a relatively long time. In addition to greater seller im-patience, a stigma effect as explained by Taylor (1999) could cause the sellingprice to be adjusted downward.

In addition, we also note that sellers who grant concessions not directly relatedto decreasing the sales price of the home also tend to sell their home for less thansellers who did not grant concessions. These concessions varied from closingcost dispensations to coverage of home inspections and warranties, agreementsto pay for certain repairs and more. All other things being equal, sellers whoagreed to concessions may have had a greater motivation to sell and thereforemay have been more impatient for a sale, which resulted in a lower selling pricefor reasons mentioned earlier.

Finally, we also note that sellers who require that their property be shownwith a prior appointment only also tended to sell their homes for less thanother sellers. Although the reason for this is unclear, one possible explanationmay be that sellers who require prior appointments may reduce the rate atwhich their property is inspected by prospective buyers. Although the parameterestimate is marginally significant in the OLS regression, 9% of the local GWRparameter estimates for prior appointment are statistically significant at the 5%significance level.

As mentioned earlier, one of the more valuable features of geographicallyweighted models is that they can produce local estimates of the parametersfor any location, even when data are not available for that location, as it usesonly the geographic coordinates for the focal location, as shown in Equation 8.A direct consequence of this feature is the ability to produce continuous surfaceplots for a particular area to understand how attribute values change as afunction of geography. Figures 1 and 2 illustrate the GWR surface plots for thecoefficients of two such attributes: total acres and log heated square feet. Eachdot in these maps represents the location of one of the houses in our sample.

From the maps shown in Figures 1 and 2, one can see that the highest valuesfor total acres appear in the southeast and, to a lesser extent, in the north tonortheastern portions of the county. Log heated square feet illustrates that thesouthwest portions of the county generally have the highest elasticities. Thus,


Figure 1 � Geographic distribution of the coefficients for acreage.

in these areas, absolute additions in acres and relative additions in heated squarefeet are likely to bring about the greatest relative increases in home prices.

Another important point of interest relates to number of days on the market.Because homes do not sell in a perfectly competitive environment, time isrequired to match sellers of this heterogeneous product with appropriate buyers(Knight 2002). Search theory is most frequently used to explain this matching


Figure 2 � Geographic distribution of the elasticities for heated square footage.

process as well as the presumed trade-off relationship between the seller’schoice of price and time on the market. However, whether a longer searchduration actually translates into a higher ultimate selling price is a subjectof controversy. As mentioned earlier, sellers who are willing to wait longerincrease the probability of encountering a buyer with a high reservation price.At the same time, however, we have also seen evidence that waiting too longcould signal that there is some possible defect or problem with the home, which


could actually lower the sales price. A chief advantage of a GWR model is thatwe can examine the changes of the local parameter estimates for time on themarket to obtain a greater understanding of the different local effects. In ourdata set, approximately 88% of the homes had positive parameter estimates fordays on the market, indicating that, for a majority of homes being sold, waitinglonger generally translates into a higher selling price. However, in 12% of thehomes being sold, we found negative parameter estimates, thereby lending atleast some evidence that, in certain areas, the potential for negative herdingexists and hence it is important for both buyers and sellers to be aware of thispossibility. Moreover, we find that 49% of these local parameter estimates fordays on the market are statistically significant at the 5% significance level.

Geographic Distribution of the Fair Market Value of a Typical Home

For a better sense of the geographical distribution of property values acrossour sample, we used our local regression parameter estimates to compute theexpected value of a typical house on a grid of locations covering the countyunder study. We took the median attributes of the houses in our sample as theattributes for the typical house:

• 2,085 heated square feet (ln(2,085) = 7.6425)

• 0 unheated square feet

• Three bedrooms

• 0.46 acres

• 13 years old in age

• 2.5 full bathrooms

• One garage

• One fireplace

• Home was on the market for 41 days

• Home was sold in the spring

• Home was occupied

• List price was not revised (original list price was used throughout salesprocess)

• Seller did not grant any concessions

• Seller did not require appointments prior to showing of home


Figure 3 � Geographic distribution of the fair market price estimate for a standardhouse.

Figure 3 displays the contour map of the fair market value of this typical housein the sampled region, confirming that there is substantial geographic variationin property values and that the same house would be worth more if located inthe southern region of the county.

While it is well known that correlations in errors can suggest omitted variablesthat have not been exploited in a model (Chatterjee, Hadi and Price 2000),


Table 4 � Average value of the standard median home by designated middle school.

Middle School Mean N Std. Dev.

Stanford $213,204 180 $18,003Stanback $248,196 136 $27,820McDougle $279,770 170 $607Smith $279,397 126 $1,263Phillips $278,174 199 $542Culbreth $281,778 224 $1,499Total $264,127 1,035 $28,581

geographically weighted models can account for these errors by allowing themto be modeled through the local parameter estimates. If this is true, then thegeographic differences in property values shown in Figure 3 should be ableto better reflect those unobservable characteristics that affect market value. Inorder to test whether this is the case, we consider an important factor knownto influence residential property values: access to good-quality education (Judand Watts 1981, Hayes and Taylor 1996, Bogart and Cromwell 1997, Crone1998, Black 1999). To test this hypothesis, we compare the average prices ofthe same standard median home shown in Figure 3 by their designated middleschool. Results from this comparison, shown in Table 4, strongly support thehypothesis; there is a clear and statistically significant difference in averageprices for the same standard median home across the designated schools, andmiddle school alone explains 81% of the variance in standard prices acrossall 1,035 locations in our sample. Similar results were obtained using thedesignated elementary schools and high schools.

The benefits of allowing for geographic variation in hedonic price analysisare numerous. As illustrated in Figure 3, it is possible to take the same homewith identical attributes and understand how the expected price changes solelyas a function of location. New housing contractors could use this method todetermine the best tracts of land to build on to maximize expected profit. It isalso possible to determine which specific housing attributes add the greatestprofits by location, and contractors could even tailor construction in differentareas to emphasize those features that are most highly valued at a given location.Following is an illustrative example.

The county used in this study had a majority of home sales in eight differentZIP codes: 27517, 27514, 27516, 27243, 27705, 27302, 27510 and 27278.Eight specific home locations (one in each of the above ZIP codes) wererandomly selected and the value for the same typical house described earlierwas computed for each of these eight locations. The “Appraiser’s Price” column


of Table 7 reflects the fair market price for the same standard (median) homeat each of these specific locations. One can see that location alone can accountfor substantial variations in this fair market price for the same standard home,ranging from $189,902 (in ZIP code 27705) to $279,831 (in ZIP code 27510).

The Buyer and Seller Perspectives: Geographically WeightedStochastic Frontier

In order to sell homes more effectively, however, more information is neededthan simply the appraiser’s estimate of home value. For example, in one area,perhaps the estimated differential between the buyer and seller reservationvalues is minimal. This might suggest that brokers have less work to do inselling properties in these areas, compared to other areas where the differentialsare large. A property with a tighter expected distribution range between buyerand seller reservation values could potentially decrease conflicts of interestbetween the seller and broker. These properties likely require less negotiationsand efforts on the parts of the broker, who can allow the properties to speak forthemselves.

The geographically weighted stochastic and inverse stochastic frontiers for thesame eight median locations are presented below. Table 5 contains the localstochastic frontiers (buyer’s reservation value) and Table 6 contains the in-verse local stochastic frontier estimates (seller’s reservation value) for thesesame eight locations. Using these estimates, it is possible to calculate predictedvalues for the maximum and minimum expected selling prices, respectively,for the identical home located in different areas; recall that, in the previ-ous example, the expected selling price (from the appraiser’s standpoint) ofa hypothetical home located in ZIP code 27510 was $279,831. Then, contin-uing with this example and using Tables 5 and 6, we see that the estimatedbuyer’s reservation value for this same hypothetical home in ZIP code 27510is $308,313, while seller’s reservation value of the same home in 27510 is$279,753.

Table 7 presents a summary of the three price estimates by location. It is inter-esting to see how the expected selling prices and ranges of price negotiationsfor the same house vary as a function of location. By looking at the location inZIP code 27243, for example, one can see that the expected appraisal estimate($244,700) is much closer to the maximum frontier of the property ($244,727)than to the minimum frontier ($212,708). This might indicate that this location(all other things being equal) is in more of a local buyer’s market, in that theestimated fair market value is close to the maximum expected price that thebuyer could expect to pay for the home. In other words, using the fair mar-ket value as a benchmark, the buyer will likely not pay much more than the


Tab

le5

�B

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ble

pric

efr

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ese

ller’

spe

rspe

ctiv

e.

Med

ian

ZIP

:Z

IP:

ZIP

:Z

IP:

ZIP

:Z

IP:

ZIP

:Z

IP:

Val

ues

2751

727

514

2751

627

243

2770

527

302

2751

027

278

Var

iabl

eH

ypot

h.L

ocal

Loc

alL

ocal

Loc

alL

ocal

Loc

alL

ocal

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alN

ame

Hom

eB

etas

Bet

asB

etas

Bet

asB

etas

Bet

asB

etas

Bet

as

Inte

rcep

t1

5.83

5.64

5.34

6.14

6.11

6.25

5.75

7.18

Log

heat

edsq

uare

feet

7.64

0.83

0.84

0.89

0.74

0.74

0.71

0.85

0.59

Log

unhe

ated

squa

refe

et0

0.01

0.01

0.01

0.01

0.01

0.00

0.01

0.01

Num

ber

ofbe

droo

ms

30.

009

−0.0

32−0

.015

0.05

30.

017

0.00

00.

001

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tala

cres

0.46

0.04

0.02

0.01

0.03

0.04

0.04

0.02

0.03

Age

ofpr

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ty(y

ears

)13

0.00

10.

004

0.00

40.

000

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001

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ber

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2.5

0.09

0.17

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0.11

0.06

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0.15

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ber

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0.03

0.04

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110.

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160.

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days

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000

the

mar

ket

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seis

vaca

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.07

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8−0

.06

−0.0

8−0

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Selle

rad

just

edlis

tpri

ce0

−0.0

2−0

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40.

05−0

.04

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3−0

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0.76

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0.86

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190.

140.

150.

110.

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120.

180.

14Pr

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for

$305

,004

$267

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$279

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$244

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$189

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$226

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$308

,313

$236

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each

loca

tion:

740 Samaha and KamakuraT

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2751

727

514

2751

627

243

2770

527

302

2751

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278

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iabl

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ocal

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asB

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asB

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as

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rcep

t1

5.73

5.88

5.21

5.58

6.11

6.29

5.60

7.11

Log

heat

edsq

uare

feet

7.64

0.83

0.78

0.89

0.80

0.73

0.70

0.86

0.60

Log

unhe

ated

squa

refe

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0.01

0.01

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ber

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015

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003

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cres

0.46

0.04

0.03

0.01

0.02

0.04

0.04

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0.03

Age

ofpr

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ty(y

ears

)13

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0.16

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000

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00.

000

mar

ket

Hou

seis

vaca

nt0

−0.0

5−0

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6−0

.02

−0.0

7−0

.07

−0.0

8−0

.08

Selle

rad

just

edlis

tpri

ce0

−0.0

2−0

.07

−0.0

40.

04−0

.04

−0.1

4−0

.03

−0.0

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ller

gran

ted

conc

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ons

0−0

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40.

000.

08−0

.05

−0.0

4Se

ller

requ

ires

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ent

00.

01−0

.06

−0.0

5−0

.05

0.00

0.05

0.00

−0.0

6δ

0.00

3.62

1.35

3.13

1.79

1.39

0.00

0.00

σ0.

160.

230.

200.

170.

160.

160.

150.

12Pr

edic

ted

selli

ngpr

ice

for

$279

,179

$223

,453

$247

,176

$212

,708

$174

,285

$205

,985

$279

,753

$221

,307

each

loca

tion:


Table 7 � Price comparisons from the appraiser’s, seller’s and buyer’s perspectives.

Buyer’s Appraiser’s Seller’s Negotiation Neg. RangeZIP Code Best Price Price Best Price Range % Appr. Price

27517 279,179 279,257 305,004 25,825 9.2%27514 223,453 267,803 267,832 44,379 16.6%27516 247,176 279,742 279,796 32,620 11.7%27243 212,708 244,700 244,727 32,020 13.1%27705 174,285 189,902 189,926 15,640 8.2%27302 205,985 226,898 226,925 20,941 9.2%27510 279,753 279,831 308,313 28,560 10.2%27278 221,307 221,356 236,783 15,476 7.0%

appraisal price, but stands to pay substantially less, particularly if the buyer isable to bargain effectively with the seller. In Figure 4 we present the range ofnegotiations (difference between the buyer and seller reservation values) acrossthe whole county, which suggests that the range of price negotiations betweenbuyers and sellers is largest in the eastern parts of the county, where propertyvalues are generally in the relatively high-priced ranges. However, this is notalways the case. It can be seen from comparing Figure 3 to Figure 4 that homeswith higher fair market valuations do not always correspond to having largestnegotiation ranges. Hence, the estimated fair market value of the home is notthe only driver of how large the potential negotiation range may be.

Application in a Bargaining Context

As mentioned earlier, bargaining power can materially affect the outcomes ofreal estate transactions. In a hedonic framework, bargaining can be thought of asa multifaceted activity, in which bargaining over the price of the home partiallyconsists of bargaining over the implicit shadow prices of each characteristic,each of which contributes to the overall bargaining outcome (Song 1995).

Our model presents a straightforward interpretation of this perspective. Eachvariable in our model has three different estimated shadow prices; two ofthe shadow prices correspond to the respective buyer and seller reservationvalues and one shadow price corresponds to the fair market value. Note thatif all the shadow prices from the buyer and seller perspectives converge tothe shadow prices for the fair market value, then the estimated reservationvalues would also converge to the fair market value and bargaining woulddissipate as homes would be expected to sell for their appraisal prices. Hence,the overall negotiation range between the buyer and seller (which defines thelatitude under which bargaining takes place) is a direct consequence of how


Figure 4 � Geographic distribution of the estimated negotiation ranges for a standardhouse.

large the differences are between the buyer’s and seller’s valuations for eachof the shadow prices. If the buyer and seller shadow prices are close to eachother for a certain housing component, then this component is likely to havelittle effect on the negotiation range, and therefore bargaining is not likelyto take place over that attribute. However, this relationship is not necessarilyconstant over geographic space. As an example, buyers and sellers may have


different shadow prices for the age of a home in an old neighborhood than inan area that is predominantly new construction. As a result, our model alsoallows for the shadow prices to vary as a function of location. Hence, in somelocations, certain housing attributes may be relatively more important in drivingthe negotiation process than in other areas. This can provide the analyst with agreater understanding of what characteristics are relatively more important inthe bargaining process as a function of location.

Figure 5 illustrates the impact that the attribute age of home can have on theoverall negotiation range. To construct Figure 5, we first set the buyer and sellerlocal parameter estimates for age equal to their respective local fair marketvalue estimates and then recalculate the negotiation range. Note that this newnegotiation range corrects for the incremental impact that age has in determininghow far apart the buyer and seller reservation values are because we have set thelocal parameter estimates for age in both cases equal to one another. We thensubtract this newly calculated negotiation range from the original negotiationrange to obtain an estimate for how the differences between the buyer’s andseller’s shadow prices for age impact the total negotiation process. Figure 5plots this change.

As illustrated in Figure 5, we see that, for a large majority of the county, thedifferences between the buyer’s and seller’s local parameter estimates for ageare positive and can increase the overall negotiation range up to approximately$7,000. However, for a small number of cases (2.9%), the sellers’ shadowprices actually exceed the buyers’ shadow prices. It is interesting to note that,while the overall reservation value for the buyer must be at least as large as theoverall reservation value for the seller (in order for a sale to occur), this does notnecessarily constrain the individual shadow prices for each attribute to also begreater for the buyer than the seller. If the seller’s shadow price for a particularattribute exceeds the buyer’s shadow price for that same attribute, then thisimplies that no negotiations occur on that attribute, which can adversely affectthe overall negotiation process if the disparity is large enough. The degree towhich this adversely affects or decreases the overall negotiation range can beseen by the locations in Figure 5 that are shaded in dark gray which occur in asmall area in the far north of the county.

Sellers and buyers, however, are likely most interested in obtaining the overallbest price for a sale. To the extent that certain markets or pockets may favorone party over another would likely be of great interest to the sellers and buyersas well as the brokers who are interested in representing them. Using the fairmarket value as a benchmark, we compute a quantity that is designed to assessthe degree to which the seller may have the upper hand in a potential sale,


Figure 5 � Geographic distribution of the change in negotiation range for a standardhouse when shadow prices for age are set to fair market value.

which is defined as follows:

ω = (Buyer.RV − Fair.Market.V alue)

(Buyer.RV − Seller.RV ). (10)

The intuition behind the above quantity is as follows: If the buyer’s reservationvalue is significantly above the fair market value, while at the same time the


seller’s reservation value is only slightly lower than the fair market value, thenthe home is likely to sell for a price that is above the fair market value, which isbeneficial to the seller. On the other hand, if the buyer’s reservation value liesclose to the fair market value but the seller’s reservation value is substantiallybelow the fair market value, then the home is likely to sell for a price thatis lower than the fair market value, which is advantageous for the buyer. Thequantity ω above is constrained to lie between 0 and 1, and a value of 0.5implies that the fair market value lies equidistant between the buyer and sellerreservation values. Values closer to 1.0 favor the seller while values closer to0.0 favor the buyer.

Figure 6 plots the quantity ω, which reflects the seller’s upside potential ofbeing able to sell the home for a price that is above the fair market value.As illustrated in Figure 6, much of the lower half of the county favors buyerswhile the upper half tends to favor sellers. There are also bands of areas (inparticular in the center of the country near the interstate highway) where buyersand sellers appear to have roughly the same upside potential for negotiating agood price, relative to the appraisal price.

The mean fair market value for the sample of the typical home across oursample of locations was approximately $264,127. Hence, the mean range ofnegotiations between the buyer and seller ($27,790) divided by the expectedfair market value sales price of $264,127 implies a 10.5% average range ofnegotiation in our sample, suggesting that homes on average tend to sell for upto about 5.25% below or above their fair market values. This result seems to beconsistent with other empirical findings regarding the effects that bargainingpower has on the selling price. For example, using data from the AmericanHousing Survey, Harding et al. (2003) find that a household’s bargaining powercan seasonally vary on the order from 5% to 7%.

Assessing the Reliability of the Frontier Estimates

In order to assess the reliability of the stochastic frontier estimates, the buyer’sreservation value can be compared to the list price. Because the list price is oftenseen as a contractual upper bound on the selling price (Horowitz 1992, Yavasand Yang 1995, Knight 2002), we should expect to see a close correspondencebetween the buyers’ estimated reservation values and the list prices for actualhomes sold. Hence, we also compute the stochastic frontiers for our original data(not the typical median home) and compare the estimated buyers’ reservationvalues to the actual list prices. Overall, we find that the buyers’ estimatedreservation values across our sample are approximately 6.4% higher than thelist price. Hence, the empirical evidence supports the proposition that, whilethe list price is relatively close to the buyer’s reservation value, the reservationvalues are slightly higher than the list prices, suggesting that reservation values


Figure 6 � Geographic distribution of the seller’s upside bargaining potential for astandard house.

do in fact represent a theoretical ideal. It should also be noted that, in oursample, 15.5% of the homes were sold for amounts greater than the list price.Hence, in a scenario where more than a small proportion of homes are sold foramounts above their theoretical contractual upper bounds, it is not surprisingto see frontiers that are slightly larger than what might be expected under moreregular conditions.


As a further check on our estimated reservation values, we also regressed thelist price against the housing characteristics and computed residuals. We thenregressed these residuals against the estimated buyer reservation values com-puted earlier and found a strong, statistically significant, positive relationshipbetween the two (p value < 0.0001). This seems to imply that, once we havecontrolled for housing characteristics, sellers with higher-than-predicted listprices are accompanied by higher buyer reservation values, while sellers withlower-than-predicted list prices are accompanied by lower buyer reservationvalues. Hence, we seem to have additional evidence that movements in thebuyer’s estimated reservation values mirror movements in the list price set bythe seller.

Conclusions and Areas for Further Research

In this article, we provide a new model that can be used to obtain a greaterunderstanding of buyer and seller reservation values as well as the fair marketvalue. Our geographically weighted stochastic frontier model provides esti-mates of the fair market value and also the maximum and minimum expectedprice for a house, given its characteristics and location, based on available MLSdata in the same real estate market. By doing so, this model provides a pricerange within which transaction prices can be negotiated, which can then beused to enhance decision-making processes along a multitude of dimensions.By estimating local frontiers using geographically weighted data, our modelalso accounts for the fact that different property features are valued differently,depending on location.

Our proposed model is useful in practical applications where there is largemarket data available but for which there is still imperfect information. For ex-ample, despite the fact that MLS data are now widely available, Yavas (1992)points out that real estate markets can best be described by imperfect informa-tion: the players do not know the locations of the reservation prices of theirpotential trading partners and that this lack of information compels each playerto engage in search activity in order to find a trading partner. In our model,however, players can now have location-specific estimates for the expectedreservation values for all parties involved. This could be used to reduce marketuncertainties and improve negotiations. Sellers who are getting ready to listtheir homes for sale could, for example, compare the suggested list price thatthe brokers recommend to the expected buyer reservation value to get a feel forhow reasonable the suggested broker list price is and also how closely alignedthe broker’s motivations are with the seller’s motivations.

This model could also be used to match potential buyers and sellers more suc-cessfully. Astute buyers, for instance, could use this information to efficiently


search for local submarkets or homes that offer lower-than-expected marketprices or search for homes where their reservation values overlap with theexpected seller reservation values.

We have also demonstrated how reservation values and fair market valuescan change across geography, which can have obvious consequences to urbanplanning, sales management and more.

As far as opportunities for future research, there are a number of areas wherethis model could be extended. One such example is to readjust the kernel toallow the anisotropic weightings to vary by location. There is no reason tobelieve that the decay factors are globally fixed throughout the study area.Using an anisotropic kernel that allows for the rates of decay to change as afunction of location might incorporate additional realism and flexibility intothe model. Another possibility lies with modeling changes in home pricestemporally as well as spatially. Hence, it would be interesting to understandhow the expected prices (and frontiers) of the median home evolve over time,in addition to location. For instance, in some areas, the reservation values maybe expected to change much more quickly than in others, indicating a widerrange of negotiations (and perhaps more market uncertainty) going forward.This might have important implications for long-term housing developmentprojects.

Extending the model to account for temporal shifts in property values wouldalso have the added benefit of increasing the sample of locations, leading tomore accurate local measures of property value. With this increased accuracy,it might be possible to extend our analyses and look at the impact of otherfactors such as proximity to highways, lakes, power plants, business districtsand so forth on property values, with obvious consequences to urban planning.

Finally, in the estimation of the buyer/seller valuations and fair market value,we used only the available market data on final transaction prices and propertyattributes, not considering the fact that some sellers may have decided not to takeany of the offers they received and some buyers may have decided not to make apurchase. In other words, we only considered cases where the buyer’s valuationwas greater than the seller’s and the final transaction price was somewhere inbetween; for this reason, our estimates are based on truncated data, because wedo not observe the cases where a compromise was not reached.

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